{"id":209,"date":"2016-08-08T22:08:03","date_gmt":"2016-08-09T06:08:03","guid":{"rendered":"http:\/\/wonghoi.humgar.com\/blog\/?page_id=209"},"modified":"2017-07-14T23:19:01","modified_gmt":"2017-07-15T07:19:01","slug":"oversimplified-signals-and-systems-4-how-to-represent-a-signal","status":"publish","type":"page","link":"https:\/\/wonghoi.humgar.com\/blog\/oversimplified-signals-and-systems-4-how-to-represent-a-signal\/","title":{"rendered":"Oversimplified: Signals and Systems (4) – How to represent a signal?"},"content":{"rendered":"<\/p>\n

A system<\/span>\u00a0can be thought of as a function that maps the input to an output. In this class, most of the systems that we are interested in are linear, therefore the functions are linear as well\u00a0so superposition<\/span> applies.<\/p>\n

\n

We haven’t defined what the inputs are. In fact it’s arbitrary, but in ‘Signals and Systems’ class, we are mostly interested in two kinds of signals (or systems): continuous time and discrete time*.<\/p>\n

For continuous time signal,\u00a0its value is\u00a0well defined over ANY\u00a0time point \"t\" you specify as long as it’s within the real line. The time can be \"3\", \"2.5\", \"\pi\", \"-\infty\", etc.\u00a0Listing the value at every time point is not an option because there are\u00a0uncountablely many<\/span>\u00a0of them. So we\u00a0describe continuous signal\u00a0as a parametric function over \"t\", e.g. \"\sin(t)\".<\/p>\n

As for discrete time signal (or time series), you can tell the value for sure ONLY at a\u00a0countable<\/span> number of time points.\u00a0Everything in between is up to interpretation, which we will study later when we get to sampling. I didn’t say these time points have to\u00a0be evenly spaced (uniformly sampled<\/span>), but they\u00a0usually are\u00a0in this class.<\/p>\n

We can\u00a0define a discrete time signal\u00a0similarly\u00a0using a\u00a0parametric function such\u00a0as \"\sin(\pi k)\", where \"k\" is the integer time index. But this time, because the number of time points in a discrete time signal is countable, you can choose to build a look-up table (a\u00a0vector<\/span>\u00a0of values, such as \"[2, 4, 5, 4, -1]^T\") if your signal is non-zero (non-trivial<\/span><\/a>) over a fixed a finite number of time points (has finite support<\/a><\/span>).<\/p>\n

In fact, vectors<\/span> (aka look-up tables)\u00a0is the representation\u00a0your computer uses to store\u00a0audio recordings: a sound clip has a finite time span and you only need a countable number of\u00a0time points to play it back\u00a0(we’ll get to that in sampling theory).\u00a0You will learn more about it when we get to Digital Signal Processing (DSP).<\/p>\n

Under the concept of vector spaces,\u00a0(parametric) functions and vectors (look-up tables) can be thought of as the same thing. All rules in linear algebra applies, which I will show you later.<\/p>\n

\n

Traditionally,\u00a0discrete time and continuous time\u00a0systems are clearly separated and taught\u00a0in any order depending on the teacher’s\u00a0preference. After you mastered the first one, the rest are almost identical with a few changes.<\/p>\n

In my opinion, it is organized like a newbie programmer copying-and-pasting code all over the place when she\/he should have refactored it into a reusable function.<\/p>\n

‘Ideas Oversimplified’ offers a new approach to learn\u00a0the essence of the\u00a04-transforms {continuous, discrete}x{Fourier, generalized}, in one shot: using\u00a0inner products in\u00a0(linear) vector spaces. More on that later.<\/p>\n

\n

* Whenever I say ‘time’ (\"t\"), it can be replaced by any parameter,\u00a0such as distance (\"x\"). It’s just a running variable. I chose not to use \"x\" other than for intro to\u00a0complex numbers because I want to reserve the alphabet \"x\" and \"y\" for input \"x(t)\" and the output \"y(t)\".<\/p>\n

<\/div>\n

<\/path><\/svg><\/i> \"Loading\"<\/p>\n

<\/div>\n","protected":false},"excerpt":{"rendered":"

A system\u00a0can be thought of as a function that maps the input to an output. In this class, most of the systems that we are interested in are linear, therefore the functions are linear as well\u00a0so superposition applies. We haven’t … Continue reading →<\/span><\/a><\/p>\n

<\/div>\n

<\/path><\/svg><\/i> \"Loading\"<\/p>\n

<\/div>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-209","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/comments?post=209"}],"version-history":[{"count":11,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/209\/revisions"}],"predecessor-version":[{"id":561,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/209\/revisions\/561"}],"wp:attachment":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/media?parent=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}